The uniform period bound conjecture for centralizers in spherical Artin–Tits groups

Let $G$ be an Artin-Tits group of spherical type. For $e004$every $e004gamma i G$, let $p$ denote the period of the sequence

$$Z(e004\gamma),Z(e004\gamma^2),Z(e004\gamma^3),\ldots.$$

Uniform period bound conjecture. There is a number $n(G)$ such that $p\leq n(G)$ for every $e004gamma i G$. This would provide a uniform bound for the centralizers needed by the canonical-reduction-system algorithm; the source gives no resolution of the conjecture.